Weakly connected Roman domination in graphs

A Roman dominating function on a graph G = (V, E) is defined to be a function f : V -> {0, 1, 2} satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. A dominating set D subset of V is a weakly connected dominating set of G if the graph (V, E boolean AND(Dx V)) is connected. We define a weakly connected Roman dominating function on a graph G to be a Roman dominating function such that the set {u is an element of V : f(u) is an element of {1, 2}} is a weakly connected dominating set of G. The weight of a weakly connected Roman dominating function is the value f(V) = Sigma(u is an element of v)f(u). The minimum weight of a weakly connected Roman dominating function on a graph G is called the weakly connected Roman domination number of G and is denoted by gamma(wc)(R) (G). In this paper, we initiate the study of this parameter. (C) 2019 Elsevier B.V. All rights reserved.